Our deterministic approximation is derived from a novel construction of the stochastic epidemic with rewiring, in which the Erdős-Rényi
network and epidemic
are constructed simultaneously. A key observation underlying this construction is that owing to symmetries in the Erdős-Rényi
network and the rewiring process it is sufficient to keep track of just the number (and not the end points) of susceptible-susceptible rewired edges. The construction yields
an infinite-dimensional continuous-time Markov chain. However the epidemic dynamics are encapsulated by the four-dimensional process which records the evolution of the numbers of susceptibles, infectives, infective-susceptible edges and susceptible-susceptible rewired edges with time. We prove
weak law of large numbers convergence of this four-dimensional process to the above-mentioned deterministic approximation using a theorem in [1]}.
